The recent evidences of presence of subsurface oceans of liquid water and ice on Saturn's moons, and the possible presence and astrobiological importance of polycyclic aromatic hydrocarbons (PAHs) in these environments, provide strong motivation for the exploration of the prebiotic chemistry in ice and to test if PAHs could be experimentally synthesized in ice surfaces under atmospheres containing methane as carbon source. In this work, we present a new design for prebiotic-chemistry experiments in ice matrix. Using this design, a mixture of products including PAHs, polar aromatic compounds, and hydrophilic acetylene-based polymers was obtained. We propose that acetylene generation in a methane/nitrogen atmosphere and subsequent polymerization to PAHs and polyynes could be a favored pathway in the presence of water freeze-melt cycles. These results shed light on the processes involved in PAH synthesis in icy environments and on the physical factors that drive the different competing pathways in methane/nitrogen atmospheres.